**Maths Class 10
Exercise 7.2**

**1. Find the coordinates of the point which
divides the join of (–1, 7) and (4, –3) in the ratio 2 : 3.**

**Solution: **Let *x*_{1} = –1, *x*_{2}
= 4, *y*_{1} = 7 and *y*_{2} = –3, *m*_{1} = 2 and *m*_{2} = 3

Let’s use the section
formula to find the coordinates of a point which divides the join of (–1, 7)
and (4, –3) in the ratio 2 : 3.

Therefore, the coordinates of the point are (1, 3) which divides the join of (–1, 7) and

(4,
–3) in the ratio 2 : 3.

**2. Find the coordinates of the points of
trisection of the line segment joining (4, –1) and (–2, –3).**

**Solution:
**

We want to find the
coordinates of the points of trisection of the line segment joining (4, –1) and
(–2, –3).

We
are given AC = CD = DB

We
want to find the coordinates of point C and D.

Let the coordinates
of the point C be (*x*_{1}, *y*_{1}) and the coordinates of
the point D be (*x*_{2}, *y*_{2}).

Clearly, the
point C divides the line segment AB in the ratio 1 : 2 and the point D divides
the line segment AB in the ratio 2 : 1.

Let’s use the section
formula to find the coordinates of point C which divides the join of A(4, –1)
and B(–2, –3) in the ratio 1 : 2.

Therefore, the
coordinates of the point C are (2, –5/3 ) and the coordinates of the point
D are (0, –7/3).

**3. To conduct Sports Day activities, in your
rectangular shaped school ground ABCD, lines have been drawn with chalk powder
at a distance of 1 m each. 100 flower pots have been placed at a distance of 1
m from each other along AD, as shown in the figure. Niharika runs ¼ th of the
distance AD on the 2nd line and posts a green flag. Preet runs 1/5 th of the
distance AD on the eighth line and posts a red flag. What is the distance
between both the flags? If Rashmi has to post a blue flag exactly halfway
between the line segment joining the two flags, where should she post her flag?**

**Solution:
**Niharika runs ¼ th
of the distance AD on the 2^{nd} line and posts a green flag.

There
are 100 flower pots between A and D. It means, she stops at 25th flower pot.

Therefore,
the coordinates of the point where she stops are (2, 25).

Preet runs 1/5 th
of the distance AD on the eighth line and posts a red flag. There are 100
flower pots between A and D. It means, she stops at 20th flower pot.

Therefore,
the coordinates of the point where she stops are (8, 20).

Let’s use the distance
formula to find the distance between points (2, 25) and (8, 20).

Rashmi posts a blue flag exactly halfway between the line segment joining the two flags.

Using
section formula to find the coordinates of this point, we get

x
= (2 + 8)/2 = 10/2 = 5

y
= (25 + 20)/2 = 45/2

Therefore,
the coordinates of the point, where Rashmi posts her flag are (5, 45/2).

It
means that she posts her flag in 5th line after covering 45/2 = 22.5 m of distance.

**4. Find the ratio in which the line segment
joining the points (–3, 10) and (6, –8) is divided by (–1, 6).**

**Solution:
**Let (–1, 6)
divides the line segment joining the points (–3, 10) and (6, –8) in the ratio *k* : 1.

Using
the section formula, we get

*k*– 1 = 6

*k*− 3

⇒ −7*k*
= −2

⇒ *k*
= 2/7

Therefore,
the ratio is 2/7 : 1 which is
equivalent to 2 : 7.

Therefore, (–1,
6) divides the line segment joining the points (–3, 10) and (6, –8) in the
ratio 2 : 7.

**5. Find the ratio in which the line segment
joining A(1, –5) and B(–4, 5) is divided by the x-axis. Also find the coordinates of the point of division.**

**Solution:
**Let the
coordinates of the point of division be (*x*,
0) and let it divides the line segment joining A(1, –5) and B(–4, 5) in the
ratio* k* : 1.

Using
the section formula, we get

–5
+ 5*k* = 0

⇒ 5*k*
= 5

⇒ *k*
= 1

Putting
the value of *k* in equation (i), we
get

Therefore, the
point (–3/2, 0) on *x*-axis divides the
line segment joining A(1, –5) and B(–4, 5) in 1 : 1.

**6. If (1, 2), (4, y), (x, 6) and (3, 5) are
the vertices of a parallelogram taken in order, find x and y.**

**Solution:
**Let A(1, 2), B(4,
*y*), C(*x*, 6) and D(3, 5) are the vertices of the parallelogram ABCD.

We know that the diagonals
of a parallelogram bisect each other. It means that the coordinates of the
midpoint of diagonal AC would be the same as the coordinates of the midpoint of
the diagonal BD. …… (i)

Using
the section formula, the coordinates of the midpoint of AC are:

Using
the section formula, the coordinates of the midpoint of BD are:

⇒ 1 + *x* = 7

⇒ *x*
= 6

Again,
according to condition (i), we also have

⇒ 8 = 5 + *y*

⇒ *y*
= 3

Therefore,
*x* = 6 and *y* = 3.

**7. Find the coordinates of a point A, where AB
is the diameter of a circle whose centre is (2, –3) and B is (1, 4).**

**Solution: **We have to find the coordinates of the
point A. We are given that AB is the diameter and the coordinates of the centre
are (2, –3) and the coordinates of the point B are (1, 4).

Let the
coordinates of the point A be (*x*, *y*).

Using the section
formula, we get

*x*+ 1

⇒ *x*
= 3

Again,
using the section formula, we get

⇒ −6 = 4 + *y*

⇒ *y*
= −10

Therefore,
the coordinates of the point A are (3, –10).

**8. If A and B are (–2, –2) and (2, –4),
respectively, find the coordinates of P such that AP = 3/7 AB and P lies on the line segment AB.**

**Solution:
**The
coordinates of points A and B are** **A(–2, –2) and B(2, –4).

It is given that AP = 3/7 AB

PB
= AB – AP = AB – 3/7 AB = 4/7 AB

So,
we have AP : PB = 3 : 4

Let
the coordinates of the point P be (*x*,
*y*).

Using
the section formula to find the coordinates of point P, we get

**9. Find the coordinates of the points which
divide the line segment joining A(–2, 2) and B(2, 8) into four equal parts.**

**Solution: **We have two points A(–2, 2) and B(2, 8).

Let P, Q and R be the points which divide the line segment AB into 4 equal parts.

Let
the coordinates of the points P(*x*_{1},
*y*_{1}), Q (*x*_{2}, *y*_{2}) and R(*x*_{3},
*y*_{3}).

We
know that AP = PQ = QR = RB.

It
means that the point P divides the line segment AB in the ratio 1 : 3.

Using
the section formula to find the coordinates of the point P, we get

Since,
AP = PQ = QR = RB.

It
means that the point Q is the midpoint of AB.

Using
the section formula to find the coordinates of the point Q, we get

Since, AP = PQ = QR = RB.

It
means that the point R divides the line segment AB in the ratio 3 : 1.

Using
the section formula to find the coordinates of the point P, we get

Therefore, P(–1, 7/2), Q(0, 5) and R(1, 13/2) are the coordinates of the points which divide the line segment AB in four equal parts.

**10. Find the area of a rhombus if its vertices
are (3, 0), (4, 5), (–1, 4) and (–2, –1) taken in order.**** [Hint: Area of a rhombus
= ½ (product of its diagonals)]**

**Solution:
**Let A(3, 0), B(4,
5), C(–1, 4) and D(–2, –1) are the vertices of the rhombus ABCD.

Using the distance
formula to find the length of the diagonal AC, we get

Since, area
of a rhombus = ½ (product of its diagonals)

=
½ × AC × BD

=
½ × 4√2 × 6√2

= 24 sq. units